Real Analysis Ontology

flow "RealAnalysisOntology" v1.0

# ============================================================
# REAL ANALYSIS ONTOLOGY — FLOW ALGEBRA
# ============================================================

SOURCE_URL := "https://kirethresearch.org/booksai-core/ontologies/calculus-analysis-ontologies/real-analysis/real-analysis-ontology.html"
DOMAIN := "Mathematical Ontologies"
CATEGORY := "Calculus and Analysis Ontologies"
SUBCATEGORY := "Real Analysis"
ENTITY := "Real Analysis"

# ---------- CORE ENTITY ----------
REAL_ANALYSIS := {
    type: MathematicalDiscipline,
    domain: Mathematics,
    parent: MathematicalAnalysis,
    purpose: "Provide a rigorous framework for the real numbers, sequences, limits, continuity, differentiation, integration, infinite series, compactness and convergence."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    REAL_NUMBER,
    ORDERED_FIELD,
    COMPLETENESS,
    SUPREMUM,
    INFIMUM,
    SEQUENCE,
    LIMIT,
    CAUCHY_SEQUENCE,
    CONTINUITY,
    UNIFORM_CONTINUITY,
    DERIVATIVE,
    RIEMANN_INTEGRAL,
    SERIES,
    UNIFORM_CONVERGENCE,
    COMPACTNESS,
    CONNECTEDNESS
}

# ---------- ONTOLOGICAL STRUCTURE ----------
REAL_ANALYSIS
    -> branch_of(MATHEMATICAL_ANALYSIS)
    -> formalises(CALCULUS_ON_REAL_NUMBERS)
    -> studies(REAL_VALUED_FUNCTIONS)
    -> studies(SEQUENCES)
    -> studies(SERIES)
    -> studies(REAL_TOPOLOGICAL_STRUCTURE)
    -> supports(FUNCTIONAL_ANALYSIS)

# ---------- REAL NUMBER SYSTEM ----------
REAL_NUMBERS
    := R

R
    -> FIELD
    -> ORDERED_FIELD
    -> COMPLETE_ORDERED_FIELD

REAL_NUMBER
    -> element_of(R)

# ---------- FIELD STRUCTURE ----------
R
    -> supports(ADDITION)
    -> supports(SUBTRACTION)
    -> supports(MULTIPLICATION)
    -> supports(DIVISION_BY_NONZERO)

FIELD_AXIOMS
    -> define(ALGEBRAIC_STRUCTURE)

# ---------- ORDER STRUCTURE ----------
for
    x,y in R

x < y
    -> ORDER_RELATION

ORDER
    -> TOTAL
    -> compatible_with(FIELD_OPERATIONS)

ORDER
    -> defines(INTERVALS)
    -> defines(UPPER_AND_LOWER_BOUNDS)

# ---------- COMPLETENESS ----------
COMPLETENESS
    -> defining_analytic_property_of(R)

LEAST_UPPER_BOUND_PROPERTY:

if
    A subset_of R
    AND A nonempty
    AND A bounded_above
then

SUPREMUM(A)
    -> exists_in(R)

COMPLETENESS
    -> supports(LIMIT_THEORY)

# ---------- SUPREMUM ----------
SUPREMUM(A)
    := least_upper_bound_of(A)

s = sup(A)

iff

for_all x in A:
    x <= s

and

for_every epsilon > 0:
    exists x in A
    such_that
    s-epsilon < x

# ---------- INFIMUM ----------
INFIMUM(A)
    := greatest_lower_bound_of(A)

INFIMUM
    -> dual_of(SUPREMUM)

inf(A)
    = -sup(-A)
       when(appropriate)

# ---------- ARCHIMEDEAN PROPERTY ----------
for_every x in R
exists n in NATURAL_NUMBERS
such_that

n > x

ARCHIMEDEAN_PROPERTY
    -> excludes(INFINITE_REAL_MAGNITUDES)

# ---------- DENSITY OF RATIONALS ----------
for
    x < y

exists
    q in RATIONAL_NUMBERS

such_that

x < q < y

RATIONALS
    -> dense_in(R)

IRRATIONALS
    -> dense_in(R)

# ---------- SEQUENCE ----------
SEQUENCE
    := mapping(
        NATURAL_NUMBERS -> R
    )

a_n
    -> nth_term

SEQUENCE
    -> ordered_infinite_collection

# ---------- SEQUENCE CONVERGENCE ----------
a_n
    -> converges_to(L)

iff

for_every epsilon > 0
exists N
such_that

n >= N
    -> abs(a_n-L) < epsilon

CONVERGENT_SEQUENCE
    -> has(UNIQUE_LIMIT)

# ---------- BOUNDED SEQUENCE ----------
a_n
    -> bounded

if
    exists M > 0
    such_that

abs(a_n) <= M
    for_all n

CONVERGENT_SEQUENCE
    -> BOUNDED

BOUNDED_SEQUENCE
    -/> necessarily(CONVERGENT)

# ---------- CAUCHY SEQUENCE ----------
a_n
    -> CAUCHY

iff

for_every epsilon > 0
exists N
such_that

m,n >= N
    -> abs(a_m-a_n) < epsilon

in R:

CAUCHY
    <-> CONVERGENT

CAUCHY_CRITERION
    -> expression_of(COMPLETENESS)

# ---------- SUBSEQUENCE ----------
SUBSEQUENCE
    := a_(n_k)

where

n_1 < n_2 < ...

if
    a_n -> L
then

every_subsequence
    -> converges_to(L)

# ---------- BOLZANO-WEIERSTRASS ----------
every_bounded_sequence_in_R^n
    -> has(CONVERGENT_SUBSEQUENCE)

BOLZANO_WEIERSTRASS
    -> compactness_related_result

# ---------- MONOTONE CONVERGENCE ----------
if
    a_n increasing
    AND bounded_above
then

a_n
    -> converges_to(sup{a_n})

if
    a_n decreasing
    AND bounded_below
then

a_n
    -> converges_to(inf{a_n})

# ---------- FUNCTION LIMIT ----------
for
    f: D -> R

lim_(x->a) f(x)
    = L

iff

for_every epsilon > 0
exists delta > 0
such_that

0 < abs(x-a) < delta
    -> abs(f(x)-L) < epsilon

# ---------- CONTINUITY ----------
f
    -> continuous_at(a)

iff

lim_(x->a) f(x)
    = f(a)

CONTINUITY
    -> preserves(NEARBY_VALUES_IN_LIMIT_SENSE)

# ---------- UNIFORM CONTINUITY ----------
f
    -> uniformly_continuous_on(D)

iff

for_every epsilon > 0
exists delta > 0
such_that

for_all x,y in D:

abs(x-y) < delta
    -> abs(f(x)-f(y)) < epsilon

delta
    -> independent_of(POINT_LOCATION)

# ---------- HEINE-CANTOR ----------
if
    K compact
    AND f continuous_on(K)
then

f
    -> uniformly_continuous_on(K)

# ---------- EXTREME VALUE THEOREM ----------
if
    K nonempty_compact
    AND f continuous_on(K)
then

exists x_min,x_max in K

such_that

f(x_min)
    <= f(x)
    <= f(x_max)

for_all x in K

# ---------- INTERMEDIATE VALUE THEOREM ----------
if
    f continuous_on([a,b])

and
    y lies_between(f(a),f(b))

then

exists c in [a,b]

such_that

f(c)=y

CONTINUITY
    -> preserves(INTERVAL_CONNECTEDNESS)

# ---------- DIFFERENTIABILITY ----------
f
    -> differentiable_at(a)

iff

f_prime(a)
    :=
lim_(h->0)
[
    f(a+h)-f(a)
]
/ h

exists_finitely

DIFFERENTIABILITY
    -> implies(CONTINUITY)

# ---------- DERIVATIVE RULES ----------
DERIVATIVE
    -> obeys(LINEARITY)
    -> obeys(PRODUCT_RULE)
    -> obeys(QUOTIENT_RULE_WHERE_DENOMINATOR_NONZERO)
    -> obeys(CHAIN_RULE)

DERIVATIVE
    -> measures(LOCAL_LINEAR_CHANGE)

# ---------- ROLLE THEOREM ----------
if
    f continuous_on([a,b])
    AND differentiable_on((a,b))
    AND f(a)=f(b)
then

exists c in (a,b)
such_that

f_prime(c)=0

# ---------- MEAN VALUE THEOREM ----------
if
    f continuous_on([a,b])
    AND differentiable_on((a,b))
then

exists c in (a,b)
such_that

f_prime(c)
    =
[f(b)-f(a)]/(b-a)

MEAN_VALUE_THEOREM
    -> links(DERIVATIVE, SECANT_SLOPE)

# ---------- RIEMANN INTEGRAL ----------
PARTITION
    := {
        a=x_0 < x_1 < ... < x_n=b
    }

TAGGED_POINTS
    := x_i_star in [x_(i-1),x_i]

RIEMANN_SUM
    :=
sum(
    f(x_i_star)
    * (x_i-x_(i-1))
)

f
    -> RIEMANN_INTEGRABLE

if
    sums_converge_to(common_limit)
    as
    mesh(PARTITION) -> 0

# ---------- INTEGRABILITY ----------
CONTINUOUS_FUNCTION_ON_CLOSED_BOUNDED_INTERVAL
    -> RIEMANN_INTEGRABLE

BOUNDED_FUNCTION_WITH_FINITE_DISCONTINUITIES
    -> RIEMANN_INTEGRABLE

RIEMANN_INTEGRABILITY
    -> depends_on(DISCONTINUITY_STRUCTURE)

# ---------- FUNDAMENTAL THEOREM OF CALCULUS ----------
if
    f continuous_on([a,b])

define

F(x)
    :=
integral_from_a_to_x(
    f(t) dt
)

then

F_prime(x)
    = f(x)

and

if
    G_prime=f
then

integral_from_a_to_b(
    f(x) dx
)
    =
G(b)-G(a)

# ---------- SERIES ----------
SERIES
    := sum_from_n_1_to_infinity(a_n)

PARTIAL_SUM_SEQUENCE
    := S_N
       = sum_from_n_1_to_N(a_n)

SERIES_CONVERGES
    iff

S_N
    -> finite_limit

# ---------- CONVERGENCE TESTS ----------
SERIES
    -> may_be_tested_by(COMPARISON_TEST)
    -> may_be_tested_by(RATIO_TEST)
    -> may_be_tested_by(ROOT_TEST)
    -> may_be_tested_by(INTEGRAL_TEST)
    -> may_be_tested_by(ALTERNATING_SERIES_TEST)

TEST_SELECTION
    -> depends_on(SERIES_STRUCTURE)

# ---------- ABSOLUTE / CONDITIONAL ----------
ABSOLUTE_CONVERGENCE
    := convergence_of(
        sum(abs(a_n))
    )

ABSOLUTE_CONVERGENCE
    -> implies(CONVERGENCE)

CONDITIONAL_CONVERGENCE
    := convergent_series
       whose_absolute_series_diverges

# ---------- FUNCTION SEQUENCES ----------
f_n
    -> sequence_of_functions

POINTWISE:
    for_each x:
        f_n(x) -> f(x)

UNIFORM:
    sup_x(
        abs(f_n(x)-f(x))
    )
    -> 0

UNIFORM_CONVERGENCE
    -> implies(POINTWISE_CONVERGENCE)

# ---------- UNIFORM LIMIT THEOREM ----------
if
    f_n continuous
    AND f_n -> f uniformly
then

f
    -> continuous

UNIFORM_CONVERGENCE
    -> preserves(CONTINUITY)

# ---------- COMPACTNESS ----------
K subset_of R^n

K compact
    iff
    K closed_and_bounded

HEINE_BOREL
    -> finite_dimensional_euclidean_characterisation

COMPACTNESS
    -> every_open_cover_has(FINITE_SUBCOVER)

# ---------- CONNECTEDNESS ----------
CONNECTED_SET
    -> cannot_be_partitioned_into(TWO_NONEMPTY_DISJOINT_OPEN_RELATIVE_SETS)

in R:

CONNECTED_SET
    <-> INTERVAL

CONNECTEDNESS
    -> explains(INTERMEDIATE_VALUE_PROPERTY)

# ---------- APPLICATIONS ----------
REAL_ANALYSIS
    -> applied_to(CALCULUS_FOUNDATIONS)
    -> applied_to(FUNCTIONAL_ANALYSIS)
    -> applied_to(PROBABILITY)
    -> applied_to(DIFFERENTIAL_EQUATIONS)
    -> applied_to(OPTIMISATION)
    -> applied_to(NUMERICAL_ANALYSIS)

# ---------- RELATION GRAPH ----------
REAL_NUMBERS
    -> have(COMPLETENESS)

COMPLETENESS
    -> supports(CAUCHY_CONVERGENCE)

SEQUENCE
    -> has(LIMIT_BEHAVIOUR)

FUNCTION_LIMIT
    -> defines(CONTINUITY)

FUNCTION_LIMIT
    -> defines(DERIVATIVE)

RIEMANN_SUM
    -> defines(INTEGRAL)

COMPACTNESS
    -> strengthens(CONTINUOUS_FUNCTION_PROPERTIES)

UNIFORM_CONVERGENCE
    -> controls(FUNCTION_LIMIT_PRESERVATION)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "real analysis" ~= "rigorous study of real-number calculus and convergence",
    "least upper bound property" ~= "order completeness of the real numbers",
    "Cauchy criterion" ~= "convergence criterion based on mutual closeness of sequence terms",
    "uniform continuity" ~= "continuity with one delta working uniformly over the domain",
    "uniform convergence" ~= "function convergence controlled uniformly across the domain",
    "Heine-Borel theorem" ~= "closed-and-bounded compactness criterion in finite-dimensional Euclidean space"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    real_completeness_is_required_for_many_convergence_results: true,
    bounded_sequence_need_not_converge_but_has_convergent_subsequence_in_finite_dimensions: true,
    differentiability_implies_continuity_but_not_conversely: true,
    uniform_continuity_stronger_than_pointwise_continuity: true,
    uniform_convergence_stronger_than_pointwise_convergence: true,
    heine_borel_closed_bounded_equivalence_is_finite_dimensional_euclidean: true,
    limit_and_integral_interchanges_require_theorem_specific_conditions: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    ontology_structure: 0.99,
    real_completeness_relation: 0.99,
    sequence_convergence_relation: 0.99,
    continuity_differentiability_relation: 0.99,
    riemann_integration_relation: 0.99,
    compactness_uniform_relation: 0.99,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(real_analysis_entity)
    -> classify(number_set_sequence_series_function_or_domain)
    -> identify(order_boundedness_and_domain)
    -> determine(limit_or_convergence_mode)
    -> select(rigorous_definition)
    -> test(relevant_theorem_hypotheses)
    -> apply(completeness_compactness_or_continuity_theorem_if_valid)
    -> analyse(differentiation_integration_or_series_if_requested)
    -> distinguish(pointwise_uniform_and_cauchy_structures)
    -> validate(all_limit_operations)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed real-analysis ontology graph",
    primary_entity: REAL_ANALYSIS,
    preserve: [
        real_number_structure,
        order,
        completeness,
        domain,
        boundedness,
        convergence_mode,
        continuity_conditions,
        differentiability_conditions
    ],
    infer_through: [
        suprema,
        infima,
        sequences,
        cauchy_sequences,
        function_limits,
        continuity,
        derivatives,
        riemann_integrals,
        series,
        compactness
    ],
    avoid: [
        assuming_bounded_sequence_converges,
        confusing_pointwise_with uniform convergence,
        claiming_continuity_implies differentiability,
        applying_heine_borel_in infinite-dimensional spaces,
        interchanging_limits_without hypotheses,
        ignoring_domain_and_endpoint conditions
    ]
}

# ---------- UPDATE MODEL ----------
UPDATE_MODEL(new_information):
    validate(new_information)
    -> identify(real_analysis_context)
    -> compare(existing_relations)
    -> preserve(completeness_order_domain_and_limit_invariants)
    -> update(affected_entities)
    -> update(affected_edges)
    -> recompute(CONFIDENCE)
    -> regenerate(FLOW_HASH)

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + REAL_ANALYSIS
    + REAL_NUMBERS
    + COMPLETENESS
    + SUPREMUM
    + SEQUENCE
    + CONTINUITY
    + DIFFERENTIABILITY
    + RIEMANN_INTEGRAL
    + SERIES
    + COMPACTNESS
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> REAL_ANALYSIS
    -> ENTITY_IDENTIFICATION
    -> STRUCTURE_CLASSIFICATION
    -> ORDER_BOUNDEDNESS_ANALYSIS
    -> LIMIT_CONVERGENCE_ANALYSIS
    -> DEFINITION_SELECTION
    -> THEOREM_HYPOTHESIS_VALIDATION
    -> COMPLETENESS_COMPACTNESS_ANALYSIS
    -> CONTINUITY_DIFFERENTIABILITY_ANALYSIS
    -> INTEGRATION_SERIES_ANALYSIS
    -> LIMIT_OPERATION_VALIDATION
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW